Control of centrifugal compressors

ABSTRACT

Surging of a centrifugal compressor is avoided by ensuring that in operation ##EQU1## WHERE K and k are parameters whose values depend on the characteristics of the compressor, g is the acceleration due to gravity, h p  is the polytropic head produced by the compressor, Vc is the velocity of sound in said inlet gas, and Mn (the Mach Number) is the ratio of the flow velocity V of the gas at the inlet to the compressor to the velocity of sound Vc therein. This is normally effected by arranging that ##EQU2## where Δ p  is the differential pressure across a throttling member disposed in an inlet duct of the compressor, P 1  is the compressor inlet pressure, and P 2  is the compressor outlet pressure.

BACKGROUND OF THE INVENTION

This invention relates to the control of centrifugal compressors toprevent surging thereof.

If the volume of gas delivered by a centrifugal compressor falls below apredetermined limit, the compressor surges. For example, if thecompressor is arranged to deliver a constant volume of air to a blastfurnace, and the varying conditions in the blast furnace causes anincrease in the resistance to the flow of the air through thecompressor, the compressor will be required to deliver to the blastfurnace a greater mass flow of air in order to maintain the said volumeof air constant at the higher discharge pressure from the compressor.If, however, sufficient air is not available at the compressor inlet,the compressor will run out of air with the result that there will be areverse flow of air through the compressor, i.e. a surge cycle willoccur. If the resistance to the flow of air through the compressor isnot then reduced, the surge cycle will be repeated until the correctvolume of air flows through the compressor.

Such surging is highly undesirable since the resultant vibration, noiseand overheating can lead to mechanical damage and ultimate wrecking ofthe compressor and of associated instrumentation and ducting connectedthereto.

The compressor must therefore be controlled to prevent surging under alloperating conditions, and this is normally achieved either byre-circulating, when necessary, a flow of the gas which has beencompressed in the compressor from the outlet to the inlet thereofthrough a by-pass duct, or by blowing off some of the gas dischargedfrom the compressor.

Precise surge control is desirable to increase the operating range ofthe compressor and to avoid unnecessary energy losses. Such precisesurge control should be responsive to changes in the composition, inletpressure and inlet temperature of the gas entering the compressor and,in many cases, should be such as to ensure that the compressor isoperated as closely as possible to the surging condition in order toobtain the best efficiency.

The conventional method of defining the surge point, i.e., theconditions in which the compressor will surge, has consisted indetermining the relationship between the outlet pressure of thecompressor and the volumetric flow through the compressor inlet. Themethod is not sufficiently accurate however since it takes no account ofvariables such as pressure, temperature, molecular weight andsupercompressability of the gas entering the compressor. Consequently,when this method is used, the compressor is liable to surge "for noapparent reason".

In an attempt to allow for some of these variables, compressormanufacturers often supply a family of curves defining surge, each suchcurve showing the said relationship between the outlet pressure and theinlet volumetric flow for predetermined conditions of inlet temperatureand pressure. Not only, however, is it difficult in practice to use sucha family of curves, but also it is by no means necessarily apparent inpractice which particular curve is applicable since the value of avariable such as the said inlet pressure may not be very accuratelyknown and does not necessarily remain constant. Consequently, it is notpracticable to operate at all close to the surge point as defined by therespective curve, and this can mean that the compressor is necessarilyvery inefficiently operated.

Various attempts have therefore been made to control a centrifugalcompressor otherwise than by merely determining the relationship betweenthe outlet pressure of the compressor and the inlet volume thereof. Forexample, in British patent specification No. 1,209,057 the compressor iscontrolled in accordance with the formula ##EQU3## where h is thepressure difference across a throttling element in the intake to thecompressor, p₁ and p₂ are respectively the inlet and outlet pressures ofthe compressor, φ and ψ are constants which depend respectively on theparticular compressor and throttling element used, and a and b areconstants which depend on the value of the compressor ratio p₂ /p₁ andon the polytropic exponent n. This formula, however, is derivedmathematically from the proposition that surging in a centrifugalcompressor depends only on the angular velocity N of the compressorrotor, whereas in fact it also depends on the temperature T, thesupercompressability Z, the ratio of the specific heats γ and themolecular weight M.W. of the inlet gas. Consequently the said formula isapplicable only to low values of the compression ratio.

SUMMARY OF THE INVENTION

According therefore to one aspect of the present invention, there isprovided apparatus comprising a centrifugal compressor; means forproducing in operation a first signal which is functionally related tothe ratio ##EQU4## where g is the acceleration due to gravity, h_(p) isthe polytropic head produced by the compressor, and Vc is the velocityof sound in said inlet gas; means for producing in operation a secondsignal which is functionally related to Mn², where Mn (the Mach Number)is the ratio of the flow velocity V of the gas at the inlet to thecompressor to the velocity of sound Vc therein; and control means,controlled by said first and second signals, for ensuring that inoperation ##EQU5## where K and k are parameters whose values depend onthe characteristics of the compressor, whereby surging of the compressoris avoided.

Preferably the means for producing the first signal is responsive to theratio P₂ /P₁, where P₁ is the compressor inlet pressure, and P₂ is thecompressor outlet pressure.

Preferably also the means for producing the second signal is responsiveto the ratio Δ_(p) /nP₁ where Δ_(p) is the differential pressure acrossa throttling member disposed in an inlet duct of the compressor, n isthe polytropic exponent of the said gas, and P₁ is the compressor inletpressure. In many cases n is a constant and may therefore for practicalpurposes be ignored.

The apparatus may comprise a duct having a control valve therein,communicates with the outlet end of the compressor, the said controlmeans controlling opening and closing of the control valve.

The said duct may, for example, be a by-pass duct which is connectedacross the compressor between the inlet and outlet ends thereof. In thiscase, the by-pass duct preferably passes through a heat exchanger sothat gas flowing from the said outlet end to the said inlet end iscooled.

Alternatively, the said duct may be a venting duct whose outlet end isopen to atmosphere.

According to another aspect of the present invention, there is provideda method for controlling a centrifugal compressor comprising producing afirst signal which is functionally related to the ratio ##EQU6## where gis the acceleration due to gravity, h_(p) is the polytropic headproduced by the compressor, and Vc is the speed of sound in said inletgas; producing a second signal which is functionally related to Mn²,where Mn (the Mach. Number) is the ratio of the flow velocity V of thegas at the inlet to the compressor to the velocity of sound Vc therein;and ensuring that ##EQU7## where K and k are parameters whose valuesdepend on the characteristics of the compressor, whereby surging of thecompressor is avoided. It may thus be arranged that ##EQU8##

It is preferably arranged that ##EQU9## where Δ_(p) is the differentialpressure across a throttling member disposed in an inlet duct of thecompressor, P₁ is the compressor inlet pressure, and P₂ is thecompressor outlet pressure.

It will thus be noted that the variables Δ_(p), P₁ and P₂ are used in atotally different way in the case of the present invention to the way inwhich similar variables are used in the case of British Pat. No.1,209,057. Thus, in the case of British Pat. No. 1,209,057 the variableP₂ is added to a function of P₁, and the ratio of Δ_(p), to thisaddition is used to control the compressor. In the case of the presentinvention, however, the compressor is controlled in functionaldependence upon the relationship of the ratio ##EQU10## to the ratio

DESCRIPTION OF THE PREFERRED EMBODIMENT

The invention is illustrated, merely by way of example in theaccompanying drawings, in which:

FIG. 1 shows a known family of curves illustrating the relationshipbetween the compressor outlet pressure P₂ and the inlet volume flow Qthrough the compressor for varying conditions,

FIG. 2 is a graph showing the relationship according to the presentinvention, between the compression ratio P₂ /P₁ and Mn², the square ofthe Mach Number of the gas entering the compressor,

FIG. 3 is a graph showing the known relationship between the polytropichead h_(p) produced by the compressor and the inlet volumetric flow Qthrough the compressor,

FIG. 4 is a graph showing the relationship according to the presentinvention between the ratio ##EQU12## and Mn²,

FIG. 5 is a graph showing the relationship according to the presentinvention between the compression ratio P₂ /P₁ and the ratio Δ_(p) /P₁,and

FIG. 6 is a schematic drawing of an apparatus according to the presentinvention.

In FIG. 1 there is shown a known family of curves illustrating therelationship between the compressor outlet pressure P₂ and the inletvolumetric flow Q through the compressor for one particular compressor.Curves of the sort shown in FIG. 1 are commonly produced by compressormanufacturers for use of their customers. As will be seen from FIG. 1,each curve relates to a specific temperature T (Winter/Summer) and aspecific compressor inlet pressure P₁ (at Altitudes A, B, C or D). Thereare thus a number of discontinuous curves which end in a surge region ona somewhat random basis, and such curves not only represent anover-simplification, in that for instance they take no account of gasmolecular weight and supercompressability, but they are also extremelydifficult to use in practice and make no allowance for varyingconditions of temperature and pressure.

The present invention is based on the discovery that if, as shown inFIG. 2, the compression ratio P₂ /P₁ is plotted against Mn² (Mn beingthe Mach Number, i.e., the ratio of the flow velocity V of the gas atthe inlet to the compressor to the velocity of sound Vc therein), thenall the information provided by the said family of curves will be givenby a single curve representing the surge line, and this single curvewill be readily usable for control purposes since it concerns therelationship between non-dimensional similarity parameters. Moreover, asindicated below, this single curve may readily be linearlised and canaccount correctly for changes in compressor inlet pressure P₁,compressor inlet temperature T, the molecular weight M.W. of the inletgas, and the ratio of the specific heats γ of the gas.

Compressor theory normally starts from incompressible fan theory inwhich the accepted non-dimensional similarity parameters used to plotthe performance of the fan are g h/N².D² and Q/ND³, where g is theacceleration due to gravity, h is the head of gas produced across thefan, N is the rotational speed of the fan, D is the diameter of the fan,and Q, as indicated above, is the inlet volumetric flow to the fan. Inthe case of the compressible flow which occurs in a centrifugalcompressor, the said head h is replaced by the polytropic head h_(p)produced by the compressor, and the value of the latter may be derivedfrom the expressions: ##EQU13## where ρ is the mass density of the saidinlet gas, n is the polytropic exponent of the compression process, andC is a constant which depends on the gas. This gives the equation##EQU14## However when preparing a graph to show the position of thesurge line it has been conventional, as shown in FIG. 3, to plot thepolytropic head, h_(p), against the inlet volumetric flow Q. This,however, is not satisfactory because the result is not non-dimensionaland because the polytropic head h_(p) cannot be measured directly.Moreover, the polytropic head h_(p) is very difficult to calculatesince, as indicated by the equation (1), it depends on the compressionratio P₂ /P₁, the polytropic exponent n, the molecular weight M.W. ofthe inlet gas, the supercompressibility Z of the gas, and the compressorinlet temperature T.

As indicated above, the Mach Number Mn is the ratio of the flow velocityV of the gas at the inlet to the compressor to the velocity of soundtherein. Thus,

    Mn=V/Vc.

The velocity of sound may be derived from the equation Vc² =dP/dρ and,for the polytropic process by the equation

    Vc.sup.2 =n P/ρ=nRTZ/G                                 (2)

where R is the gas constant, and G is the specific gravity of the inletgas.

Consequently the equation (2) can be used to non-dimensionalise thesurge line graph shown in FIG. 3, in which the polytropic head h_(p) isplotted against the inlet volumetric flow Q, to give that shown in FIG.4, in which the ratio ##EQU15## is plotted against ##EQU16## where A isthe inlet area of the compressor.

The area above the surge line shown in FIG. 4 is the area in whichsurging will occur. Consequently, if surging is to be avoided, ##EQU17##where K and k are parameters dependent on the shape of the surge lineand are thus parameters whose values depend on the characteristics ofthe compressor. These parameters K and k can be easily and exactlydetermined in practice by plotting the surge line on the axes shown inFIG. 4 either by using information provided by the compressormanufacturer for the benefit of his customers or by obtaining suchinformation from the results of conventional experiments.

As will be appreciated from the above, ##EQU18##

It can be seen that ##EQU19## is a weak function of n, but is a strongfunction of the compression ratio P₂ /P₁.

Therefore we may write as an approximation ##EQU20##

If a throttling member is disposed in the intake to the compressor, thedifferential pressure Δp across the throttling member is in accordancewith the expression Δp∝ρV². Thus by using the equation Vc² =nP/ρ ofequation (2), we obtain ##EQU21##

Furthermore, if n is almost constant, this simplifies to

    Mn.sup.2  Δp/P.sub.1.

thus a very good approximation to compressor performance and surgecontrol would be given by the graph shown in FIG. 5 where thecompression ratio P₂ /P₁ is plotted against the ratio Δp/P₁. In thiscase surge control can be effected merely by measuring the variables P₁,P₂, and Δp, as in the schematic embodiment shown in FIG. 6.

If n is not a constant, it may be treated as a function of G, thespecific gravity of the gas. For example, for natural gasn=1.4727-0.280G. G itself can be measured directly by a specific gravitymeter or calculated from the expression ##EQU22##

In FIG. 6 there is shown a centrifugal compressor 10 having an inletduct 11 and an outlet duct 12. The inlet duct 11 and outlet duct 12 haverespective flow valves 13, 14 therein. A by-pass duct 15, having aby-pass valve 16 therein, is connected across the compressor 10 betweenthe inlet and outlet ends thereof and communicates with the inlet duct11 and outlet duct 12. The by-pass duct 15 preferably passes as shownthrough a heat-exchanger 17 so that a by-pass flow of gas flowingthrough the by-pass duct 15 from the outlet end to the inlet end of thecompressor is cooled in passing through the heat exchanger 17.

Disposed in the inlet duct 11 is a throttling member 20 the differentialpressure Δp across which is measured by a transducer 21. The inletpressure P₁ to the compressor 10, i.e., downstream of the throttlingmember 20, is measured, by a transducer 22, while the outlet pressure P₂from the compressor is measured by a transducer 23.

A control means 24, which controls opening and closing of the by-passvalve 16, comprises a divider 25 which receives signals from thetransducers 21, 22. The divider 25 produces an output signal which isdependent upon the ratio Δp/P₁ and which is passed to an analogue ordigital computer 26. Thus the output signal from the divider 25 isfunctionally related to Mn².

The control means 24 also comprises a divider 27 which receives signalsfrom the transducers 22, 23. The divider 27 produces an output signalwhich is dependent upon the ratio P₂ /P₁ and which is passed to thecomputer 26. Thus the output signal from the divider 27 is functionallyrelated to the ratio ##EQU23##

The computer 26 compares the values of ##EQU24## with pre-programmedinformation and provided that ##EQU25## the by-pass valve 16 ismaintained closed. However if ##EQU26## a signal is passed to a two modecontroller 30 which opens the by-pass valve 16. Thus surging is avoided.

Alternatively, the by-pass valve 16 may be pneumatically operated, inwhich case a current to pneumatic converter 31 is interposed between thetwo mode controller 30 and the by-pass valve 16.

If desired, the duct 15, instead of being a by-pass duct, could be aventing duct whose inlet end communicates with the outlet end of thecompressor 10 the venting duct 15 having an outlet end 32 which is opento atmosphere.

We claim:
 1. Apparatus comprising a centrifugal compressor; means forproducing in operation a first signal which is functionally related tothe ratio ##EQU27## where g is the acceleration due to gravity, h_(p) isthe polytropic head produced by the compressor, and Vc is the velocityof sound in inlet gas entering the compressor; means for producing inoperation a second signal which is functionally related to Mn², where Mn(the Mach Number) is the ratio of the flow velocity V of the gas at theinlet to the compressor to the velocity of sound Vc therein; and controlmeans for preventing surging of the compressor, said control means beingcontrolled by said first and second signals, and ensuring that inoperation ##EQU28## where K and k are parameters whose values depend onthe characteristics of the compressor.
 2. Apparatus as claimed in claim1 in which the means for producing the first signal is responsive to theratio P₂ /P₁, where P₁ is the compressor inlet pressure and P₂ is thecompressor outlet pressure.
 3. Apparatus as claimed in claim 1 in whichthe means for producing the second signal is responsive to the ratioΔp/n P₁ where Δp is the differential pressure across a throttling memberdisposed in an inlet duct of the compressor, n is the polytropicexponent of the said gas, and P₁ is the compressor inlet pressure. 4.Apparatus as claimed in claim 3 in which n is a constant.
 5. Apparatuscomprising a centrifugal compressor; means for producing in operation afirst signal which is functionally related to the ratio ##EQU29## whereg is the acceleration due to gravity, h_(p) is the polytropic headproduced by the compressor, and Vc is the velocity of sound in inlet gasentering the compressor; means for producing in operation a secondsignal which is functionally related to Mn², where Mn (the Mach Number)is the ratio of the flow velocity V of the gas at the inlet of thecompressor to the velocity of sound Vc therein; control means which arecontrolled by said first and second signals and which ensure that inoperation ##EQU30## where K and k are parameters whose values depend onthe characteristics of the compressor, and a duct, having a controlvalve therein, which communicates with the outlet end of the compressor,the said control means controlling opening and closing of the controlvalve, whereby surging of the compressor is avoided.
 6. Apparatus asclaimed in claim 5 in which the said duct is a by-pass duct which isconnected across the compressor between the inlet and outlet endsthereof.
 7. Apparatus as claimed in claim 6 in which the by-pass ductpasses through a heat exchanger so that gas flowing from the said outletend to the said inlet end is cooled.
 8. Apparatus as claimed in claim 5in which the said duct is a venting duct whose outlet end is open toatmosphere.
 9. A method of controlling a centrifugal compressionproducing a first signal which is functionally related to the ratio##EQU31## where g is the acceleration due to gravity, h_(p) is thepolytropic head produced by the compressor, and Vc is the speed of soundin inlet gas entering the compressor; producing a second signal which isfunctionally related to Mn², where Mn (the Mach Number) is the ratio ofthe flow velocity V of the gas at the inlet to the compressor to thevelocity of sound Vc therein; and employing said first and secondsignals to prevent surging of the compressor by ensuring that ##EQU32##where K and k are parameters whose values depend on the characteristicsof the compressor.
 10. A method as claimed in claim 9 in which it isarranged that ##EQU33##
 11. A method as claimed in claim 9 in which itis arranged that ##EQU34## where Δ_(p) is the differential pressureacross a throttling member disposed in an inlet duct of the compressor,P₁ is the compressor inlet pressure, and P₂ is the compressor outletpressure.